Deriving the Value of an Exponent

[UMich1344]

Homework Statement

Suppose we lived in a matter-dominated, critical-density universe. The matter density rho of such a universe changes with time t as a power of t. Derive the value of the exponent n in the following equation:

Homework Equations

rho = At^n

where A = constant

 

[Sourabh N]

Show your attempt first.

 

[ogb p]

In a matter-dominated, critical-density universe, the matter density is expected to change with time t as a power of t. This can be mathematically represented as rho = At^n, where A is a constant. To derive the value of the exponent n, we can use the fact that in a critical-density universe, the density of matter is equal to the critical density, which is given by the critical density parameter, Ωm. This parameter is defined as the ratio of the actual matter density to the critical density.

Therefore, we can write the equation as Ωm = At^n / ρcrit, where ρcrit is the critical density. Since we are assuming a matter-dominated universe, we can also use the fact that the matter density is proportional to the scale factor a(t) raised to the power of -3. This can be written as ρm ∝ a(t)^-3.

Substituting this into the equation above, we get Ωm = A a(t)^-3 t^n / ρcrit. Rearranging this equation and taking the logarithm on both sides, we get ln(Ωm) = -3ln(a(t)) + nln(t) + ln(A/ρcrit).

Now, if we take the time derivative of both sides, we get dln(Ωm)/dt = -3dln(a(t))/dt + n/t + 0. Since we are in a critical-density universe, the value of Ωm is constant over time, so the left side of the equation is equal to 0. Thus, we get 0 = -3dln(a(t))/dt + n/t. Rearranging this, we get n/t = 3dln(a(t))/dt.

Finally, we can integrate both sides with respect to t to get n = 3ln(a(t)) + C, where C is a constant of integration. Since we are assuming a matter-dominated universe, the scale factor a(t) is proportional to the age of the universe t, so we can write a(t) ∝ t^(2/3). Substituting this into the equation, we get n = 3ln(t^(2/3)) + C = 2ln(t) + C. Therefore, the value of the exponent n is 2, with a constant C that depends on the initial conditions of the universe.